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  "body": "\n## Definition\n\nLet $n \\ge 1$, let $I$ be an open interval, let $A : I \\to \\mathbb{R}^{n \\times n}$ be a continuous matrix-valued function and let $b : I \\to \\mathbb{R}^n$ be continuous.\nThe system of linear differential equations with coefficient matrix $A$ and right side $b$ is the equation\n$$x'(t) = A(t)\\, x(t) + b(t)$$\nfor an unknown differentiable function $x : I \\to \\mathbb{R}^n$.\nThe system is homogeneous if $b = 0$ on $I$.\n\n## Remarks\n\nIn coordinates, it is the system $x_i' = \\sum_{j=1}^{n} a_{ij}(t)\\, x_j + b_i(t)$ for $i = 1, \\dots, n$.\nFor each $t_0 \\in I$ and $x_0 \\in \\mathbb{R}^n$, there is exactly one solution on all of $I$ with $x(t_0) = x_0$.\nThe solutions of the homogeneous system form a vector space of dimension $n$. A basis gives a [[fundamental matrix|0MQ24RBG93]].\nA [[linear equation of order n|0PB7K8M7FG]] becomes such a system by [[reduction to a first-order system|0TH9CPW3VS]].\n\n## Explanations\n\n### L1 human\n\nSeveral unknown functions change at the same time, and each rate is a linear combination of all the unknowns.\nIn matrix form, the system looks like one linear first-order equation, with a matrix in place of a number.\n",
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